Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.

Let d > 0 be a squarefree integer and denote by h = h(−d) the class number of the imaginary quadratic field . It is well known (see e.g. [25]) that for a given positive integer N there are only finitely many squarefree d's for which h(−d) = N. In [45], Saradha and Srinivasan and in [28] Le and Zhu considered the equation in the title and solved it completely under the assumption h(−d) = 1 apart from the case d ≡ 7 (mod 8) in which case y was supposed to be odd. We investigate the title equation in unknown integers (x, y, l, n) with x ≥ 1, y ≥ 1, n ≥ 3, l ≥ 0 and gcd(x, y) = 1. The purpose of this paper is to extend the above result of Saradha and Srinivasan to the case h(−d) ∈ {2, 3}.

We prove a Noether–Deuring theorem for the derived category of bounded complexes of modules over a Noetherian algebra.

The covers of cyclic acts over monoids were investigated by Mahmoudi and Renshaw (M. Mahmoudi and J. Renshaw, On covers of cyclic acts over monoids, Semigroup Forum77 (2008), 325–338) and the authors posed some open problems. In the present paper, we give answers to their problems 1 and 5, and we also give a sufficient and necessary condition that a cyclic act has a weakly pullback flat cover.

Dimer models are a combinatorial tool to describe certain algebras that appear as noncommutative crepant resolutions of toric Gorenstein singularities. Unfortunately, not every dimer model gives rise to a noncommutative crepant resolution. Several notions of consistency have been introduced to deal with this problem. In this paper, we study the major different notions in detail and show that for dimer models on a torus, they are all equivalent.

Let μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

We generalise statements known about Springer fibres associated to nilpotents with two Jordan blocks to Spaltenstein varieties. We study the geometry of generalised irreducible components (i.e. Bialynicki-Birula cells) and their pairwise intersections. In particular, we develop a graphical calculus that encodes their structure as iterated fibre bundles with ℂℙ1 as base spaces, and compute their cohomology. At the end, we present a connection with coloured cobordisms generalising the construction of Khovanov (M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (2000), 359–426) and Stroppel (C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Mathematica145(4) (2009), 954–992).

Sather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

In this corrigendum to the paper, “Goldie*-supplemented modules” [1], we present revised Propositions 3.7 and 3.12 and correct some typographical errors.

We study an (n + 1)-tensor norm αr extending to (n + 1)-fold tensor products, the classical one of Lapresté in the case n = 1. We characterise the maps of the minimal and the maximal multi-linear operator ideals related to αr in the sense of Defant and Floret (A. Defant and K. Floret, Tensor norms and operator ideals, North Holland Mathematical Studies, no. 176 (North Holland, Amsterdam, Netherlands, 1993). As an application we give a complete description of the reflexivity of the αr-tensor product (⊗j = 1n + 1 ℓuj, αr).

A left R-module M is called two-degree Gorenstein flat if there exists an exact sequence of Gorenstein flat left R-modules ⋅⋅⋅ → G2 → G1 → G0 → G−1 → G−2 → ⋅⋅⋅ such that M ≅ Ker(G0 → G−1) and it remains exact after applying H ⊗R- for any Gorenstein injective right R-module H. In this paper we first give some characterisations of Gorenstein flat objects in the category of complexes of modules and then use them to show that two notions of the two-degree Gorenstein flat and the Gorenstein flat left R-modules coincide when R is right coherent.

We investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials. We prove that for a finite ring if the factor by the Jacobson radical can be lifted in the centre, then this problem can be solved in polynomial time. This result provides a step in proving a dichotomy conjecture of Lawrence and Willard (J. Lawrence and R. Willard, The complexity of solving polynomial equations over finite rings (manuscript, 1997)).

For an integer n ≥ 7, let Δ(n) denote the (2, 3, n)-triangle group, and let M(n) be the positive integer determined by the conditions that Δ(n) has a subgroup of index m for all m ≥ M(n), but no subgroup of index M(n) − 1. The main purpose of the paper is to obtain information (bounds, in some cases explicit values) concerning the function M(n) (cf. Theorem 1). We also show that Δ(n) is replete (i.e., has a subgroup of index m for every integer m ≥ 1) if, and only if, n is divisible by 20 or by 30 (see Theorem 2).

Let p be a prime number, Qp the field of p-adic numbers, K a finite field extension of Qp, K a fixed algebraic closure of K and Cp the completion of K with respect to the p-adic valuation. We introduce and investigate an equivalence relation on Cp, defined in terms of field extensions and metric properties of Galois orbits over K.

In this paper, we deal with complete hypersurfaces immersed with bounded higher order mean curvatures in steady state-type spacetimes and in hyperbolic-type spaces. By applying a generalised maximum principle for the Yau's square operator [11], we obtain uniqueness results in each of these ambient spaces.

The aim of this paper is to extend for the m-quasi-Einstein metrics some integral formulae obtained in [1] (C. Aquino, A. Barros and E. Ribeiro Jr., Some applications of the Hodge-de Rham decomposition to Ricci solitons, Results Math. 60 (2011), 245–254) for Ricci solitons and derive similar results achieved there. Moreover, we shall extend the m-Bakry-Emery Ricci tensor for a vector field X on a Riemannian manifold instead of a gradient field ∇f, in order to obtain some results concerning these manifolds that generalize their correspondents to a gradient field.

We study the existence of positive solutions for equations of the form where 0 < ω < π, subject to various non-local boundary conditions defined in terms of the Riemann–Stieltjes integrals. We prove the existence and multiplicity of positive solutions for these boundary value problems in both resonant and non-resonant cases. We discuss the resonant case by making a shift and considering an equivalent non-resonant problem.